I know that λ={λ} is true I want to know How can I prove that λ={λ} is true or not? Thanks in advance.
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How do you know this is true? – Julien Mar 05 '13 at 13:23
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1It is rather unclear what you are asking. Although it looks somewhat similar to this question from set theory: When is $x={x}$?. But your question is tagged [tag:automata], so you have probably something completely different in mind. – Martin Sleziak Mar 05 '13 at 13:24
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In my reference book i read about it and the writer said its equal – emma Mar 05 '13 at 13:25
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@ Martin Sleziak: the lambda is empty set – emma Mar 05 '13 at 13:26
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Could you perhaps tell us what $\lambda$ is and the name of the book (and link, if the text is online)? – Martin Sleziak Mar 05 '13 at 13:26
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@ Martin Sleziak: no its not an online book. its an automata question – emma Mar 05 '13 at 13:28
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1Are you sure $\lambda$ refers to the empty set, and not the null string? Of course, even if it does refer to the null string, I can't guess that they mean by the equation. (e.g. what is ${ \lambda }$?) – Mar 05 '13 at 13:35
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1@emma: Martin asked for the name of the book, not whether it was online or not. – hmakholm left over Monica Mar 05 '13 at 15:02
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Martin is right, assuming his interpretation of the question.
However, it looks more likely to me that you've encountered a common abuse of notation in formal language theory, in which any word can be used as a shorthand for the language containing that word and nothing else.
This is convenient -- but potentially confusing -- because among other things it allows regular expressions to be written fairly succinctly. For example $ab^*cb$ means the language $\{acb,abcb,abbcb,abbbbcb,\ldots\}$. Without this convention we'd need to be write the regular expression as $\{a\}\{b\}^*\{cb\}$, wasting ink and attention on brackets that don't really add useful meaning in most contexts.
Your $\lambda=\{\lambda\}$ is then just this convention in the special case where the word is the empty word. It doesn't really say that $\lambda$ is its own singleton set; just that we're traditionally allowed to write $\lambda$ where we really mean $\{\lambda\}$. The onus is then on the reader to figure out that must be what we meant because $\{\lambda\}$ makes better sense than $\lambda$ in the context we use it in.
(Though, of course, the real onus is on the writer not to use such abbreviation in contexts where there is any doubt about their meaning).
hmakholm left over Monica
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1+1 I think this is the relevant answer in this context, given the tags on the question. – Tara B Mar 05 '13 at 21:52
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You wrote in your comment that $\lambda$ is your notation for the empty set. A more standard notation is $\emptyset$, so I will use this notation.
What you wrote is not true, $\emptyset\ne\{\emptyset\}$.
Two sets $A$ and $B$ are equal if they contain the same elements, i.e., if $$(\forall x)x\in A\Leftrightarrow x\in B.$$ This is called extensionality.
This equivalence does not hold for $A=\emptyset$ and $B=\{\emptyset\}$. Just take $x=\emptyset$. You have $\emptyset\in B$ and $\emptyset\notin A$.
In other words, $\emptyset$ belongs to $\{\emptyset\}$, but it does not belong to $\emptyset$, therefore $\{\emptyset\}$ and $\emptyset$ must be different sets.
Martin Sleziak
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This question is very similar to yours (identical, if I understand your question correctly): Empty set does not belong to empty set – Martin Sleziak Mar 05 '13 at 13:39